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// Copyright 1995-2016 The OpenSSL Project Authors. All Rights Reserved.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
// https://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.
#include <openssl/bn.h>
#include <openssl/err.h>
#include <openssl/mem.h>
#include "../../internal.h"
#include "internal.h"
// kPrimes contains the first 1024 primes.
static const uint16_t kPrimes[] = {
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89,
97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151,
157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223,
227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281,
283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359,
367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433,
439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503,
509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593,
599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659,
661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743,
751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827,
829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911,
919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997,
1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069,
1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163,
1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249,
1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321,
1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439,
1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511,
1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601,
1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693,
1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783,
1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877,
1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987,
1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069,
2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143,
2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267,
2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347,
2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423,
2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543,
2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657,
2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713,
2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801,
2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903,
2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011,
3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119,
3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221,
3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323,
3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413,
3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527,
3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607,
3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697,
3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797,
3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907,
3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003,
4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093,
4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211,
4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283,
4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409,
4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513,
4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621,
4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721,
4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813,
4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937,
4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011,
5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113,
5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233,
5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351,
5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443,
5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531,
5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653,
5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743,
5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849,
5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939,
5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073,
6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173,
6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271,
6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359,
6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473,
6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581,
6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701,
6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803,
6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907,
6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997,
7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121,
7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229,
7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349,
7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487,
7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561,
7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669,
7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757,
7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879,
7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009,
8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111,
8117, 8123, 8147, 8161,
};
// BN_prime_checks_for_size returns the number of Miller-Rabin iterations
// necessary for generating a 'bits'-bit candidate prime.
//
//
// This table is generated using the algorithm of FIPS PUB 186-4
// Digital Signature Standard (DSS), section F.1, page 117.
// (https://doi.org/10.6028/NIST.FIPS.186-4)
// The following magma script was used to generate the output:
// securitybits:=125;
// k:=1024;
// for t:=1 to 65 do
// for M:=3 to Floor(2*Sqrt(k-1)-1) do
// S:=0;
// // Sum over m
// for m:=3 to M do
// s:=0;
// // Sum over j
// for j:=2 to m do
// s+:=(RealField(32)!2)^-(j+(k-1)/j);
// end for;
// S+:=2^(m-(m-1)*t)*s;
// end for;
// A:=2^(k-2-M*t);
// B:=8*(Pi(RealField(32))^2-6)/3*2^(k-2)*S;
// pkt:=2.00743*Log(2)*k*2^-k*(A+B);
// seclevel:=Floor(-Log(2,pkt));
// if seclevel ge securitybits then
// printf "k: %5o, security: %o bits (t: %o, M: %o)\n",k,seclevel,t,M;
// break;
// end if;
// end for;
// if seclevel ge securitybits then break; end if;
// end for;
//
// It can be run online at: http://magma.maths.usyd.edu.au/calc
// And will output:
// k: 1024, security: 129 bits (t: 6, M: 23)
// k is the number of bits of the prime, securitybits is the level we want to
// reach.
// prime length | RSA key size | # MR tests | security level
// -------------+--------------|------------+---------------
// (b) >= 6394 | >= 12788 | 3 | 256 bit
// (b) >= 3747 | >= 7494 | 3 | 192 bit
// (b) >= 1345 | >= 2690 | 4 | 128 bit
// (b) >= 1080 | >= 2160 | 5 | 128 bit
// (b) >= 852 | >= 1704 | 5 | 112 bit
// (b) >= 476 | >= 952 | 5 | 80 bit
// (b) >= 400 | >= 800 | 6 | 80 bit
// (b) >= 347 | >= 694 | 7 | 80 bit
// (b) >= 308 | >= 616 | 8 | 80 bit
// (b) >= 55 | >= 110 | 27 | 64 bit
// (b) >= 6 | >= 12 | 34 | 64 bit
static int BN_prime_checks_for_size(int bits) {
if (bits >= 3747) {
return 3;
}
if (bits >= 1345) {
return 4;
}
if (bits >= 476) {
return 5;
}
if (bits >= 400) {
return 6;
}
if (bits >= 347) {
return 7;
}
if (bits >= 308) {
return 8;
}
if (bits >= 55) {
return 27;
}
return 34;
}
// num_trial_division_primes returns the number of primes to try with trial
// division before using more expensive checks. For larger numbers, the value
// of excluding a candidate with trial division is larger.
static size_t num_trial_division_primes(const BIGNUM *n) {
if (n->width * BN_BITS2 > 1024) {
return OPENSSL_ARRAY_SIZE(kPrimes);
}
return OPENSSL_ARRAY_SIZE(kPrimes) / 2;
}
// BN_PRIME_CHECKS_BLINDED is the iteration count for blinding the constant-time
// primality test. See |BN_primality_test| for details. This number is selected
// so that, for a candidate N-bit RSA prime, picking |BN_PRIME_CHECKS_BLINDED|
// random N-bit numbers will have at least |BN_prime_checks_for_size(N)| values
// in range with high probability.
//
// The following Python script computes the blinding factor needed for the
// corresponding iteration count.
/*
import math
# We choose candidate RSA primes between sqrt(2)/2 * 2^N and 2^N and select
# witnesses by generating random N-bit numbers. Thus the probability of
# selecting one in range is at least sqrt(2)/2.
p = math.sqrt(2) / 2
# Target around 2^-8 probability of the blinding being insufficient given that
# key generation is a one-time, noisy operation.
epsilon = 2**-8
def choose(a, b):
r = 1
for i in xrange(b):
r *= a - i
r /= (i + 1)
return r
def failure_rate(min_uniform, iterations):
""" Returns the probability that, for |iterations| candidate witnesses, fewer
than |min_uniform| of them will be uniform. """
prob = 0.0
for i in xrange(min_uniform):
prob += (choose(iterations, i) *
p**i * (1-p)**(iterations - i))
return prob
for min_uniform in (3, 4, 5, 6, 8, 13, 19, 28):
# Find the smallest number of iterations under the target failure rate.
iterations = min_uniform
while True:
prob = failure_rate(min_uniform, iterations)
if prob < epsilon:
print min_uniform, iterations, prob
break
iterations += 1
Output:
3 9 0.00368894873911
4 11 0.00363319494662
5 13 0.00336215573898
6 15 0.00300145783158
8 19 0.00225214119331
13 27 0.00385610026955
19 38 0.0021410539126
28 52 0.00325405801769
16 iterations suffices for 400-bit primes and larger (6 uniform samples needed),
which is already well below the minimum acceptable key size for RSA.
*/
#define BN_PRIME_CHECKS_BLINDED 16
static int probable_prime(BIGNUM *rnd, int bits);
static int probable_prime_dh(BIGNUM *rnd, int bits, const BIGNUM *add,
const BIGNUM *rem, BN_CTX *ctx);
static int probable_prime_dh_safe(BIGNUM *rnd, int bits, const BIGNUM *add,
const BIGNUM *rem, BN_CTX *ctx);
BN_GENCB *BN_GENCB_new(void) {
return reinterpret_cast<BN_GENCB *>(OPENSSL_zalloc(sizeof(BN_GENCB)));
}
void BN_GENCB_free(BN_GENCB *callback) { OPENSSL_free(callback); }
void BN_GENCB_set(BN_GENCB *callback,
int (*f)(int event, int n, struct bn_gencb_st *), void *arg) {
callback->callback = f;
callback->arg = arg;
}
int BN_GENCB_call(BN_GENCB *callback, int event, int n) {
if (!callback) {
return 1;
}
return callback->callback(event, n, callback);
}
void *BN_GENCB_get_arg(const BN_GENCB *callback) { return callback->arg; }
int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe, const BIGNUM *add,
const BIGNUM *rem, BN_GENCB *cb) {
BIGNUM *t;
int i, j, c1 = 0;
int checks = BN_prime_checks_for_size(bits);
if (bits < 2) {
// There are no prime numbers this small.
OPENSSL_PUT_ERROR(BN, BN_R_BITS_TOO_SMALL);
return 0;
} else if (bits == 2 && safe) {
// The smallest safe prime (7) is three bits.
OPENSSL_PUT_ERROR(BN, BN_R_BITS_TOO_SMALL);
return 0;
}
bssl::UniquePtr<BN_CTX> ctx(BN_CTX_new());
if (ctx == nullptr) {
return 0;
}
bssl::BN_CTXScope scope(ctx.get());
t = BN_CTX_get(ctx.get());
if (!t) {
return 0;
}
loop:
// make a random number and set the top and bottom bits
if (add == NULL) {
if (!probable_prime(ret, bits)) {
return 0;
}
} else {
if (safe) {
if (!probable_prime_dh_safe(ret, bits, add, rem, ctx.get())) {
return 0;
}
} else {
if (!probable_prime_dh(ret, bits, add, rem, ctx.get())) {
return 0;
}
}
}
if (!BN_GENCB_call(cb, BN_GENCB_GENERATED, c1++)) {
// aborted
return 0;
}
if (!safe) {
i = BN_is_prime_fasttest_ex(ret, checks, ctx.get(), 0, cb);
if (i == -1) {
return 0;
} else if (i == 0) {
goto loop;
}
} else {
// for "safe prime" generation, check that (p-1)/2 is prime. Since a prime
// is odd, We just need to divide by 2
if (!BN_rshift1(t, ret)) {
return 0;
}
// Interleave |ret| and |t|'s primality tests to avoid paying the full
// iteration count on |ret| only to quickly discover |t| is composite.
//
// TODO(davidben): This doesn't quite work because an iteration count of 1
// still runs the blinding mechanism.
for (i = 0; i < checks; i++) {
j = BN_is_prime_fasttest_ex(ret, 1, ctx.get(), 0, NULL);
if (j == -1) {
return 0;
} else if (j == 0) {
goto loop;
}
j = BN_is_prime_fasttest_ex(t, 1, ctx.get(), 0, NULL);
if (j == -1) {
return 0;
} else if (j == 0) {
goto loop;
}
if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, i)) {
return 0;
}
// We have a safe prime test pass
}
}
// we have a prime :-)
return 1;
}
static int bn_trial_division(uint16_t *out, const BIGNUM *bn) {
const size_t num_primes = num_trial_division_primes(bn);
for (size_t i = 1; i < num_primes; i++) {
// During RSA key generation, |bn| may be secret, but only if |bn| was
// prime, so it is safe to leak failed trial divisions.
if (constant_time_declassify_int(bn_mod_u16_consttime(bn, kPrimes[i]) ==
0)) {
*out = kPrimes[i];
return 1;
}
}
return 0;
}
int bn_odd_number_is_obviously_composite(const BIGNUM *bn) {
uint16_t prime;
return bn_trial_division(&prime, bn) && !BN_is_word(bn, prime);
}
int bn_miller_rabin_init(BN_MILLER_RABIN *miller_rabin, const BN_MONT_CTX *mont,
BN_CTX *ctx) {
// This function corresponds to steps 1 through 3 of FIPS 186-4, C.3.1.
const BIGNUM *w = &mont->N;
// Note we do not call |BN_CTX_start| in this function. We intentionally
// allocate values in the containing scope so they outlive this function.
miller_rabin->w1 = BN_CTX_get(ctx);
miller_rabin->m = BN_CTX_get(ctx);
miller_rabin->one_mont = BN_CTX_get(ctx);
miller_rabin->w1_mont = BN_CTX_get(ctx);
if (miller_rabin->w1 == NULL || //
miller_rabin->m == NULL || //
miller_rabin->one_mont == NULL || //
miller_rabin->w1_mont == NULL) {
return 0;
}
// See FIPS 186-4, C.3.1, steps 1 through 3.
if (!bn_usub_consttime(miller_rabin->w1, w, BN_value_one())) {
return 0;
}
miller_rabin->a = BN_count_low_zero_bits(miller_rabin->w1);
if (!bn_rshift_secret_shift(miller_rabin->m, miller_rabin->w1,
miller_rabin->a, ctx)) {
return 0;
}
miller_rabin->w_bits = BN_num_bits(w);
// Precompute some values in Montgomery form.
if (!bn_one_to_montgomery(miller_rabin->one_mont, mont, ctx) ||
// w - 1 is -1 mod w, so we can compute it in the Montgomery domain, -R,
// with a subtraction. (|one_mont| cannot be zero.)
!bn_usub_consttime(miller_rabin->w1_mont, w, miller_rabin->one_mont)) {
return 0;
}
return 1;
}
int bn_miller_rabin_iteration(const BN_MILLER_RABIN *miller_rabin,
int *out_is_possibly_prime, const BIGNUM *b,
const BN_MONT_CTX *mont, BN_CTX *ctx) {
// This function corresponds to steps 4.3 through 4.5 of FIPS 186-4, C.3.1.
bssl::BN_CTXScope scope(ctx);
// Step 4.3. We use Montgomery-encoding for better performance and to avoid
// timing leaks.
const BIGNUM *w = &mont->N;
BIGNUM *z = BN_CTX_get(ctx);
crypto_word_t is_possibly_prime;
if (z == NULL ||
!BN_mod_exp_mont_consttime(z, b, miller_rabin->m, w, ctx, mont) ||
!BN_to_montgomery(z, z, mont, ctx)) {
return 0;
}
// is_possibly_prime is all ones if we have determined |b| is not a composite
// witness for |w|. This is equivalent to going to step 4.7 in the original
// algorithm. To avoid timing leaks, we run the algorithm to the end for prime
// inputs.
is_possibly_prime = 0;
// Step 4.4. If z = 1 or z = w-1, b is not a composite witness and w is still
// possibly prime.
is_possibly_prime = BN_equal_consttime(z, miller_rabin->one_mont) |
BN_equal_consttime(z, miller_rabin->w1_mont);
is_possibly_prime = 0 - is_possibly_prime; // Make it all zeros or all ones.
// Step 4.5.
//
// To avoid leaking |a|, we run the loop to |w_bits| and mask off all
// iterations once |j| = |a|.
for (int j = 1; j < miller_rabin->w_bits; j++) {
if (constant_time_declassify_w(constant_time_eq_int(j, miller_rabin->a) &
~is_possibly_prime)) {
// If the loop is done and we haven't seen z = 1 or z = w-1 yet, the
// value is composite and we can break in variable time.
break;
}
// Step 4.5.1.
if (!BN_mod_mul_montgomery(z, z, z, mont, ctx)) {
return 0;
}
// Step 4.5.2. If z = w-1 and the loop is not done, this is not a composite
// witness.
crypto_word_t z_is_w1_mont = BN_equal_consttime(z, miller_rabin->w1_mont);
z_is_w1_mont = 0 - z_is_w1_mont; // Make it all zeros or all ones.
is_possibly_prime |= z_is_w1_mont; // Go to step 4.7 if |z_is_w1_mont|.
// Step 4.5.3. If z = 1 and the loop is not done, the previous value of z
// was not -1. There are no non-trivial square roots of 1 modulo a prime, so
// w is composite and we may exit in variable time.
if (constant_time_declassify_w(
BN_equal_consttime(z, miller_rabin->one_mont) &
~is_possibly_prime)) {
break;
}
}
*out_is_possibly_prime = constant_time_declassify_w(is_possibly_prime) & 1;
return 1;
}
int BN_primality_test(int *out_is_probably_prime, const BIGNUM *w, int checks,
BN_CTX *ctx, int do_trial_division, BN_GENCB *cb) {
// This function's secrecy and performance requirements come from RSA key
// generation. We generate RSA keys by selecting two large, secret primes with
// rejection sampling.
//
// We thus treat |w| as secret if turns out to be a large prime. However, if
// |w| is composite, we treat this and |w| itself as public. (Conversely, if
// |w| is prime, that it is prime is public. Only the value is secret.) This
// is fine for RSA key generation, but note it is important that we use
// rejection sampling, with each candidate prime chosen independently. This
// would not work for, e.g., an algorithm which looked for primes in
// consecutive integers. These assumptions allow us to discard composites
// quickly. We additionally treat |w| as public when it is a small prime to
// simplify trial decryption and some edge cases.
//
// One RSA key generation will call this function on exactly two primes and
// many more composites. The overall cost is a combination of several factors:
//
// 1. Checking if |w| is divisible by a small prime is much faster than
// learning it is composite by Miller-Rabin (see below for details on that
// cost). Trial division by p saves 1/p of Miller-Rabin calls, so this is
// worthwhile until p exceeds the ratio of the two costs.
//
// 2. For a random (i.e. non-adversarial) candidate large prime and candidate
// witness, the probability of false witness is very low. (This is why FIPS
// 186-4 only requires a few iterations.) Thus composites not discarded by
// trial decryption, in practice, cost one Miller-Rabin iteration. Only the
// two actual primes cost the full iteration count.
//
// 3. A Miller-Rabin iteration is a modular exponentiation plus |a| additional
// modular squares, where |a| is the number of factors of two in |w-1|. |a|
// is likely small (the distribution falls exponentially), but it is also
// potentially secret, so we loop up to its log(w) upper bound when |w| is
// prime. When |w| is composite, we break early, so only two calls pay this
// cost. (Note that all calls pay the modular exponentiation which is,
// itself, log(w) modular multiplications and squares.)
//
// 4. While there are only two prime calls, they multiplicatively pay the full
// costs of (2) and (3).
//
// 5. After the primes are chosen, RSA keys derive some values from the
// primes, but this cost is negligible in comparison.
*out_is_probably_prime = 0;
if (BN_cmp(w, BN_value_one()) <= 0) {
return 1;
}
if (!BN_is_odd(w)) {
// The only even prime is two.
*out_is_probably_prime = BN_is_word(w, 2);
return 1;
}
// Miller-Rabin does not work for three.
if (BN_is_word(w, 3)) {
*out_is_probably_prime = 1;
return 1;
}
if (do_trial_division) {
// Perform additional trial division checks to discard small primes.
uint16_t prime;
if (bn_trial_division(&prime, w)) {
*out_is_probably_prime = BN_is_word(w, prime);
return 1;
}
if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, -1)) {
return 0;
}
}
if (checks == BN_prime_checks_for_generation) {
checks = BN_prime_checks_for_size(BN_num_bits(w));
}
bssl::UniquePtr<BN_CTX> new_ctx;
if (ctx == nullptr) {
new_ctx.reset(BN_CTX_new());
if (new_ctx == nullptr) {
return 0;
}
ctx = new_ctx.get();
}
// See C.3.1 from FIPS 186-4.
bssl::BN_CTXScope scope(ctx);
BIGNUM *b = BN_CTX_get(ctx);
bssl::UniquePtr<BN_MONT_CTX> mont(BN_MONT_CTX_new_consttime(w, ctx));
BN_MILLER_RABIN miller_rabin;
crypto_word_t uniform_iterations = 0;
if (b == nullptr || mont == nullptr ||
// Steps 1-3.
!bn_miller_rabin_init(&miller_rabin, mont.get(), ctx)) {
return 0;
}
// The following loop performs in inner iteration of the Miller-Rabin
// Primality test (Step 4).
//
// The algorithm as specified in FIPS 186-4 leaks information on |w|, the RSA
// private key. Instead, we run through each iteration unconditionally,
// performing modular multiplications, masking off any effects to behave
// equivalently to the specified algorithm.
//
// We also blind the number of values of |b| we try. Steps 4.1–4.2 say to
// discard out-of-range values. To avoid leaking information on |w|, we use
// |bn_rand_secret_range| which, rather than discarding bad values, adjusts
// them to be in range. Though not uniformly selected, these adjusted values
// are still usable as Miller-Rabin checks.
//
// Miller-Rabin is already probabilistic, so we could reach the desired
// confidence levels by just suitably increasing the iteration count. However,
// to align with FIPS 186-4, we use a more pessimal analysis: we do not count
// the non-uniform values towards the iteration count. As a result, this
// function is more complex and has more timing risk than necessary.
//
// We count both total iterations and uniform ones and iterate until we've
// reached at least |BN_PRIME_CHECKS_BLINDED| and |iterations|, respectively.
// If the latter is large enough, it will be the limiting factor with high
// probability and we won't leak information.
//
// Note this blinding does not impact most calls when picking primes because
// composites are rejected early. Only the two secret primes see extra work.
// Using |constant_time_lt_w| seems to prevent the compiler from optimizing
// this into two jumps.
for (int i = 1; constant_time_declassify_w(
(i <= BN_PRIME_CHECKS_BLINDED) |
constant_time_lt_w(uniform_iterations, checks));
i++) {
// Step 4.1-4.2
int is_uniform;
if (!bn_rand_secret_range(b, &is_uniform, 2, miller_rabin.w1)) {
return 0;
}
uniform_iterations += is_uniform;
// Steps 4.3-4.5
int is_possibly_prime = 0;
if (!bn_miller_rabin_iteration(&miller_rabin, &is_possibly_prime, b,
mont.get(), ctx)) {
return 0;
}
if (!is_possibly_prime) {
// Step 4.6. We did not see z = w-1 before z = 1, so w must be composite.
*out_is_probably_prime = 0;
return 1;
}
// Step 4.7
if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, i - 1)) {
return 0;
}
}
declassify_assert(uniform_iterations >= (crypto_word_t)checks);
*out_is_probably_prime = 1;
return 1;
}
int BN_is_prime_ex(const BIGNUM *candidate, int checks, BN_CTX *ctx,
BN_GENCB *cb) {
return BN_is_prime_fasttest_ex(candidate, checks, ctx, 0, cb);
}
int BN_is_prime_fasttest_ex(const BIGNUM *a, int checks, BN_CTX *ctx,
int do_trial_division, BN_GENCB *cb) {
int is_probably_prime;
if (!BN_primality_test(&is_probably_prime, a, checks, ctx, do_trial_division,
cb)) {
return -1;
}
return is_probably_prime;
}
int BN_enhanced_miller_rabin_primality_test(
enum bn_primality_result_t *out_result, const BIGNUM *w, int checks,
BN_CTX *ctx, BN_GENCB *cb) {
// Enhanced Miller-Rabin is only valid on odd integers greater than 3.
if (!BN_is_odd(w) || BN_cmp_word(w, 3) <= 0) {
OPENSSL_PUT_ERROR(BN, BN_R_INVALID_INPUT);
return 0;
}
if (checks == BN_prime_checks_for_generation) {
checks = BN_prime_checks_for_size(BN_num_bits(w));
}
bssl::BN_CTXScope scope(ctx);
BIGNUM *w1 = BN_CTX_get(ctx);
if (w1 == nullptr || !BN_copy(w1, w) || !BN_sub_word(w1, 1)) {
return 0;
}
// Write w1 as m*2^a (Steps 1 and 2).
int a = 0;
while (!BN_is_bit_set(w1, a)) {
a++;
}
BIGNUM *m = BN_CTX_get(ctx);
if (m == nullptr || !BN_rshift(m, w1, a)) {
return 0;
}
BIGNUM *b = BN_CTX_get(ctx);
BIGNUM *g = BN_CTX_get(ctx);
BIGNUM *z = BN_CTX_get(ctx);
BIGNUM *x = BN_CTX_get(ctx);
BIGNUM *x1 = BN_CTX_get(ctx);
if (b == nullptr || g == nullptr || z == nullptr || x == nullptr ||
x1 == nullptr) {
return 0;
}
// Montgomery setup for computations mod w
bssl::UniquePtr<BN_MONT_CTX> mont(BN_MONT_CTX_new_for_modulus(w, ctx));
if (mont == nullptr) {
return 0;
}
// The following loop performs in inner iteration of the Enhanced Miller-Rabin
// Primality test (Step 4).
for (int i = 1; i <= checks; i++) {
// Step 4.1-4.2
if (!BN_rand_range_ex(b, 2, w1)) {
return 0;
}
// Step 4.3-4.4
if (!BN_gcd(g, b, w, ctx)) {
return 0;
}
if (BN_cmp_word(g, 1) > 0) {
*out_result = bn_composite;
return 1;
}
// Step 4.5
if (!BN_mod_exp_mont(z, b, m, w, ctx, mont.get())) {
return 0;
}
// Step 4.6
if (BN_is_one(z) || BN_cmp(z, w1) == 0) {
goto loop;
}
// Step 4.7
for (int j = 1; j < a; j++) {
if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) {
return 0;
}
if (BN_cmp(z, w1) == 0) {
goto loop;
}
if (BN_is_one(z)) {
goto composite;
}
}
// Step 4.8-4.9
if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) {
return 0;
}
// Step 4.10-4.11
if (!BN_is_one(z) && !BN_copy(x, z)) {
return 0;
}
composite:
// Step 4.12-4.14
if (!BN_copy(x1, x) || !BN_sub_word(x1, 1) || !BN_gcd(g, x1, w, ctx)) {
return 0;
}
if (BN_cmp_word(g, 1) > 0) {
*out_result = bn_composite;
} else {
*out_result = bn_non_prime_power_composite;
}
return 1;
loop:
// Step 4.15
if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, i - 1)) {
return 0;
}
}
*out_result = bn_probably_prime;
return 1;
}
static int probable_prime(BIGNUM *rnd, int bits) {
do {
if (!BN_rand(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD)) {
return 0;
}
} while (bn_odd_number_is_obviously_composite(rnd));
return 1;
}
static int probable_prime_dh(BIGNUM *rnd, int bits, const BIGNUM *add,
const BIGNUM *rem, BN_CTX *ctx) {
bssl::BN_CTXScope scope(ctx);
BIGNUM *t1;
if ((t1 = BN_CTX_get(ctx)) == nullptr) {
return 0;
}
if (!BN_rand(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD)) {
return 0;
}
// we need ((rnd-rem) % add) == 0
if (!BN_mod(t1, rnd, add, ctx)) {
return 0;
}
if (!BN_sub(rnd, rnd, t1)) {
return 0;
}
if (rem == nullptr) {
if (!BN_add_word(rnd, 1)) {
return 0;
}
} else {
if (!BN_add(rnd, rnd, rem)) {
return 0;
}
}
// we now have a random number 'rand' to test.
size_t num_primes = num_trial_division_primes(rnd);
loop:
for (size_t i = 1; i < num_primes; i++) {
// check that rnd is a prime
if (bn_mod_u16_consttime(rnd, kPrimes[i]) <= 1) {
if (!BN_add(rnd, rnd, add)) {
return 0;
}
goto loop;
}
}
return 1;
}
static int probable_prime_dh_safe(BIGNUM *p, int bits, const BIGNUM *padd,
const BIGNUM *rem, BN_CTX *ctx) {
bits--;
bssl::BN_CTXScope scope(ctx);
BIGNUM *t1 = BN_CTX_get(ctx);
BIGNUM *q = BN_CTX_get(ctx);
BIGNUM *qadd = BN_CTX_get(ctx);
if (qadd == NULL) {
return 0;
}
if (!BN_rshift1(qadd, padd)) {
return 0;
}
if (!BN_rand(q, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD)) {
return 0;
}
// we need ((rnd-rem) % add) == 0
if (!BN_mod(t1, q, qadd, ctx)) {
return 0;
}
if (!BN_sub(q, q, t1)) {
return 0;
}
if (rem == NULL) {
if (!BN_add_word(q, 1)) {
return 0;
}
} else {
if (!BN_rshift1(t1, rem)) {
return 0;
}
if (!BN_add(q, q, t1)) {
return 0;
}
}
// we now have a random number 'rand' to test.
if (!BN_lshift1(p, q)) {
return 0;
}
if (!BN_add_word(p, 1)) {
return 0;
}
size_t num_primes = num_trial_division_primes(p);
loop:
for (size_t i = 1; i < num_primes; i++) {
// check that p and q are prime
// check that for p and q
// gcd(p-1,primes) == 1 (except for 2)
if (bn_mod_u16_consttime(p, kPrimes[i]) == 0 ||
bn_mod_u16_consttime(q, kPrimes[i]) == 0) {
if (!BN_add(p, p, padd)) {
return 0;
}
if (!BN_add(q, q, qadd)) {
return 0;
}
goto loop;
}
}
return 1;
}